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. As a model for nonlinear approximation from a redundant set of time-frequency atoms, we consider approximation in L 2 (IR) with linear combinations of Walsh at oms. Best approximation can be realized with a fast algorithm when the class of approximants is restricted to linear combinations of pairwise orthogonal atoms. We describe the effect of this restriction on approximation rates, and then discuss the performance of the greedy algorithm. In particular, a uniform geometric rate of convergence is shown to hold for the class of initial functions consisting of linear combinations of two atoms. x1. Introduction Given a dictionary D = fe g 2 of elements in a Hilbert space H the nonlinear approximation error of a given element f 2 H relative to D is E n (f) = inf g2\Sigma n kf \Gamma gk; (1:1) where \Sigma n denotes the set of linear combinations of n dictionary elements from D. If we have an algorithm that produces n term approximations to f , a fundamental question is then how ...

Abstract—We address the best basis problem—or, more generally, the best representation problem: Given a signal, a dictionary of representations, and an additive cost function, the aim is to select the representation from the dictionary which minimizes the cost for the given signal. We develop a new framework of multitree dictionaries, which includes some previously proposed dictionaries as special cases. We show how to efficiently find the best representation in a multitree dictionary using a recursive tree-pruning algorithm. We illustrate our framework through several examples, including a novel block image coder, which significantly outperforms both the standard JPEG and quadtree-based methods and is comparable to embedded coders such as JPEG2000 and SPIHT. Index Terms—Best basis, grammar, image compression, JPEG. I.

Abstract—This paper proposes, analyzes, and illustrates several best basis search algorithms for dictionaries consisting of lapped orthogonal bases. It improves upon the best local cosine basis selection based on a dyadic tree [10], [11] by considering larger dictionaries of bases. It is shown that this can result in sparser representations and approximate shift invariance. An algorithm that is strictly shift invariant is also provided. The experiments in this paper suggest that the new dictionaries can be advantageous for time-frequency analysis, compression, and noise removal. Accelerated versions of the basic algorithm are provided that explore various tradeoffs between computational efficiency and adaptability. It is shown that the proposed algorithms are in fact applicable to any finite dictionary comprised of lapped orthogonal bases. One such novel dictionary is proposed that constructs the best local cosine representation in the frequency domain, and it is shown that the new dictionary is better suited for representing certain types of signals. Index Terms—Best basis, lapped transforms, time-frequency analysis. I.

Noiselets are functions which are noise-like in the sense that they are totally uncompressible by orthogonal wavelet packet methods. We describe a library of such functions and demonstrate a few of their noise-like properties. © 2001 Academic Press As the reader undoubtedly knows, various effective algorithms exist for using wavelets and wavelet packets to process data, for example, for compression or noise removal. In these algorithms, analysis of data is achieved because one is able to find rapid decay in the distribution of values of the data, when it is transformed into wavelet or wavelet packet